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(11) | EP 0 114 938 A2 |
| (12) | EUROPEAN PATENT APPLICATION |
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| (54) | On-the-fly multibyte error correction |
| (57) In the multibyte error correcting system, up to t errors are correctable by processing
2t syndrome bytes. Syndrome bytes are converted into t+1 coefficients of the error
locator polynomial by predetermined product operations and exclusive-OR operations
involving selected syndrome bytes to produce cofactors that correspond to the desired
coefficients when less then t errors occurred in the codeword. The cofactors are combined
to produce coefficients when t errors occur and the correct set of coefficients are
selected in accordance with the number of errors that are detected. Up to t erroneous
bytes in one codeword and corrected during its transfer from the system while the
next codeword is entered in the system, so that correction is on-the-fly. |
Background of the Invention
Field of the Invention
[0001] This invention relates in general to error correction and, in the preferred embodiment to an error correcting system for correcting multibyte errors in a codeword in which the identity of the error locations and the identity of the error patterns are achieved simultaneously.
Cross-Referenced Application
[0002] Reference is made to Application serial number (US 454392) (IBM) , entitled "Syndrome Processing for Multibyte Error Correcting Systems", Applicant's reference SA982029A), which may be employed in a system of the present invention.
Description of the Prior Art
[0003] Most data storage subsystems associated with modern information handling systems employ some type of error correction system in order to obtain cost effective design for high reliability and data integrity. The ability of the data processing system to retrieve data from the storage system, i.e., access time, is a well recognized measure of the efficiency of the overall storage system. In most data processing systems, the decoding time for the error correction code is a direct factor in the total access time. As the capacity of storage devices has increased, the need for increased reliability and availability has also increased. As a result, the time required to process soft errors by the error correcting system becomes a larger percentage of the total access time. Multibyte error correcting systems suggested in the prior art require a relatively long time for decoding of the multibyte errors. This has been one of the main objections to their use in high performance storage systems.
[0004] The following references disclose basic and significant aspects of prior art error correcting systems:
1. I. S. Reed and G. Solomon, "Polynomial Codes Over Certain Finite Fields", J. Siam, 8 (1960) p. 300-304.
2. R. C. Bose and D. K. Ray-Chaudhuri, "On a Class of Error Correcting Binary Group Codes", Information and Control, 3 (1960) p. 68-79.
3. A Hocquenghem, "Codes Correcteurs d'erreurs", Chiffres (Paris) 2 (1959) p. 147-156.
4. W. W. Peterson, "Encoding and Error Correction Procedures for the Bose-Chaudhuri Codes", IEEE Transaction Information Theory, 6 (1960) p. 459-470.
5. D. C. Gorenstein and N. Zierler, "A Class of Error-Correcting Codes in pm Symbols", Journal of Soc. Indus. Applied Math. 9 (1961) p. 207-214.
6. E. R. Berlekamp, "On Decoding Binary Bose-Chaudhuri-Hocquenghem Codes", IEEE Trans. Info. Theory 11 (1965) p. 577-579.
7. J. L. Massey, "Step-by-Step Decoding of the Bose-Chaudhuri-Hocquenghem Codes", IEEE Trans. Info. Theory 11 (1965) p. 580-585.
8. R. T. Chien, "Cyclic Decoding Procedure for the Bose-Chaudhuri-Hocquenghem Codes", IEEE Trans. Info. Theory 10 (1964) p. 357-363.
9. G. D. Forney, Jr., "On Decoding BCH Codes", IEEE Trans. Info. Theory 11 (1965) p. 549-557.
10. W. W. Peterson and E. J. Weldon, Jr., Error Correcting Codes, 2nd Edition (MIT Press, 1972).
[0005] References 1, 2 and 3 provide a wide class of cyclic error correcting codes which are commonly known as Reed-Solomon Codes and BCH Codes. These codes, when defined over nonbinary or extended binary finite fields, can be used for correction of symbol errors as binary byte errors.
[0006] References 4 and 5 provide the basic key to the solution of the multi-error decoding problem by suggesting the use of the "error locator polynomial". These references (4 and 5) suggest the use of a set of linear equations to solve for the coefficients of the error locator polynomial.
[0007] References 6 and 7 suggest the use of an interative method to compute the coefficients of the error locator polynomial. The roots of the error locator polynomial represent the locations of the symbols in error.
[0008] Reference 8 suggests a simple mechanized method, the "Chien Search", for searching these roots using a cyclic trial and error procedure, while reference 9 provides further simplification in the computation of error values in the case of codes with non-binary or higher order binary symbols.
[0009] The contribution of the error locator polynomial in references 4 and 5 was significant in that the roots of this polynomial represent the locations of the symbols in error for a multibyte error correcting system. The cross-referenced copending application discloses an improved syndrome processing unit for developing the coefficients of the error locator polynomial.
[0010] The method of decoding multibyte errors in a multibyte error correcting system generally comprises four sequential steps:
Step 1 - calculating of the error syndromes.
Step 2 - determination of the coefficients of the error locator polynomial from the error syndromes.
Step 3 - identifying the error locations from the error locator polynomial by "Chien Search".
Step 4 - determination of the byte error values for each of the error locations.
[0011] All known methods for decoding multiple errors require completion of step 3 before beginning step 4. Because of that situation, prior art systems also require additional hardware to accumulate the results of step 3 for use subsequently by the hardware that implements step 4. References 6 and 10 provide good discussion on the prior art decoding methods.
Summary of the Invention
[0012] The present invention is directed to an error correcting system in which the problems of the prior art have been eliminated.
[0013] In accordance with the present invention, an error correcting system is provided in which steps 3 and 4 of the decoding procedure for multibyte errors are completed simultaneously. In particular, the location and value of each error are computed in cyclic order without the explicit information regarding the locations of the remaining errors which are yet to be determined. Step 3, which involves the "Chien Search" function is expanded into a complete mechanization of the error correcting procedure, and as a result, data characters in the codeword can be transferred from the error correcting system to the data processing system one at a time in synchronism with each cycle of the Chien Search. The access time to the first byte of data is, therefore, not impacted by the time required for the identification of the error locations and development of the error values for the errors. The hardware of the error correcting system is also highly simplified since the results of step 3 which normally had to be stored no longer need to be stored. The same set of hardware obtains the error values and the correction for all errors at the appropriate cycle during the Chien Search even when the actual number of errors is less than the designed maximum. Such an arrangement was only possible in prior art single symbol correcting codes.
[0014] The improved system also avoids all division operations in obtaining the coefficients of the error location equation when the syndrome processing unit disclosed in the copending application is employed in connection with the error pattern processing circuits of the present invention.
[0015] As discussed in the copending application, the situation where less than the maximum number of errors are present in the code word results in additional simplification of the hardware which becomes important when the circuitry of the error correcting system is being implemented in very large scale integration. The simplification is important in that when the circuitry has been implemented in VLSI to operate with 2t input syndrome bytes, the same VLSI circuitry can be used in environments which provide less than 2t input syndrome bytes.
[0016] It is, therefore, an object of the present invention to provide a multibyte error correcting system which has a minimal effect on the access time of an associated data storage system.
[0017] A further object of the present invention is to provide a multibyte error correcting system in which the transfer of the codeword from the error correcting system is initiated prior to the time that the identity of all the error locations is known.
[0018] A still further object of the present invention is to provide a multibyte error correcting system in which the location and error pattern of each byte of the codeword that is in error is simultaneously identified as bytes are being transferred from the error correcting system so that errors are corrected on-the-fly.
[0019] A further object of the present invention is to provide an ECC system which is operable to correct any number of errors up to the maximum number for which it has been designed so that the same hardware may be employed to correct less than the maximum number of errors in one application or, alternately, employed with a different application which employs a fewer number of check bytes and syndrome bytes.
[0020] A further object of the present invention is to provide an ECC system in which only one inverse operation is involved in developing the error patterns.
[0021] The foregoing and other objects, features and advantages of the invention will be apparent from the following more particular description of a preferred embodiment of the invention as illustrated in the accompanying drawing.
Brief Description of the Drawing
[0022]
FIG. 1 is a block diagram of a multibyte error correcting system embodying the present invention;
FIGS. 2 and 3 are schematic diagrams of the syndrome processing unit shown in FIG. 1 for developing the error locator coefficients of the error locator polynomial;
FIG. 4 is a schematic logic diagram of the logic for generating the coefficients for the error value expression;
FIG. 5 is a schematic diagram of the circuitry for implementing the error search, determination of the error value, and delivery of data with on-the-fly correction of the erroneous bytes.
Description of the Preferred Embodiment
[0023] A description of the system shown in FIG. 1 will first be provided. The description of the syndrome processing logic for identifying error locations and the method of operating the logic will then be followed by a mathematical explanation and proof of the manner in which the logic has been implemented. This explanation will describe in mathematical terms the operation of the error correcting system for the general case of any number of errors.
[0024] FIG. 1 shows the block diagram of an on-the-fly ECC system, which includes a syndrome processing logic disclosed and claimed in our previously cross-referenced application. As described in that application, the syndrome processing unit developes the coefficients for the error locator equation. The correcting process of the FIG. 1 system is continuous. An uninterrupted stream of data enters and leaves the decoder in the form of a chain of n-symbol codewords, hence the name, on-the-fly decoding.
[0025] From a practical viewpoint, a given decoding process can be considered on-the-fly if it meets the following test, namely, the corrected data bytes of a previously received codeword are delivered to the user system while the data bytes of the following codeword are being received.
[0026] The decoder, comprising blocks 6, 7, 8, 9, computes syndromes for the incoming codeword as it decodes and corrects errors in the previously received outgoing codeword. Each clock cycle corresponds to an input of one data symbol of the incoming codeword concurrent with an output of one corrected data symbol of the outgoing codeword. A buffer 5 holds at least n-symbols of the uncorrected data in between the incoming and outgoing symbols.
[0027] A three-error correcting Reed-Solomon Code in GF(28) is used as an example of special interest for applications in computer products. The 256 elements of GF(28) are conventionally represented by the set of 8-bit binary vectors. One such representation is given in Table 1. In a three-error correcting Reed-Solomon code, there are six check symbols corresponding to the roots α0, α1, α2, α3, α4, α5 of the generator polynomial where is an element of a finite field GF(28) and is represented by an 8-bit binary vector. The corresponding syndromes computed by block 6 are denoted by S0, S1, S2, S3, S4, and S5 respectively. These syndromes are computed from the received codeword in the conventional manner in accordance with any known prior art process. The implementation for this step is well known and makes use of exclusive-OR circuits and shift registers. The details of the block 7 logic circuits are shown in FIGS. 2 and 3.
[0029] 0114938 The overall function of the block 7, shown in FIGS. 2 and 3, is first to implement the following four equations in FIG. 2 to develop the locator parameters Δ33' Δ32, Δ31 and A30 for the three error case, which also includes the parameters Δ22' Δ21 and Δ20 for the two error case where, for example, the subscript 33 identifies one of four parameters. From these locator parameters, the coefficients Δ3' Δ2' Δ1 and Δ0 are selected by the logic shown in FIG. 3 in accordance with the exact number of errors that are involved in the particular codeword. The equations for Δ33, Δ32' Δ31 and Δ30 are as follows:
[0030] These parameters are used to determine the coefficients of the error locator equation (17). The error locations and error patterns are then determined by the blocks 8 and 9 of the "on-the-fly" system shown in FIG. 1. The details of block 7 are shown in FIG. 2 and 3.
[0031] The combinatorial logic shown in FIG. 2 includes two basic logic blocks 10 and 11. The first block 10, represented by an X, corresponds to a product operation in GF(2 )involving two 8-bit binary vectors, while the second block 11 represents a sum operation in GF(28) involving two 8-bit binary vectors. The operation of block 11 is a simple bit-by-bit exclusive-OR logical function using eight 2-way exclusive-OR gates. The product operation, on the other hand, represented by block 10 is more complex and involves 71 exclusive-OR circuits and 64 AND circuits. The need for the 71 XOR circuits and 64 AND circuits may be seen from the following explanation of the product function of block 10.
[0032] The product operation of block 10 involves two 8-bit vectors A and B to produce a third vector C where
[0033]
The product is obtained through a two step process. First, compute the coefficients fi of the product polynomial F where F = A x B, modulo 2. Computation of the coefficients fi (i = 0, ... 14) requires 64 AND gates and 49 EX-OR gates:
[0034] Second, reduce the polynomial F, modulo p(x), where p(x) is a primitive binary polynomial of degree 8. Use p(x) = 1 + x3 + x5 + x7 + x8. The reduction of fi modulo p(x) requires at the most 22 EX-OR gates.
[0035] The implementation of the product process involves one 2-input AND-gate for each product term required for the coefficient f0 through f14 and a 2-input exclusive-OR gate for combining the outputs of the AND-gates. Each block 10, therefore, represents 64 AND-gates, 71 exclusive-OR gates.
[0036] The first term of the error locator polynomial S2(S1, S3 ⊕ S2, S2) of the Δ33 equation is implemented by dashed block 16 in FIG. 2. The output of block 16 is exclusive-ORed in gate 18 with the second term of the equation and the result exclusive-ORed in gate 19 with the last term of the equation.
[0037] The blocks involved in developing each of the other parameters Δ32' Δ31 and Δ30 may be traced in a similar manner in FIG. 2. '
[0038] The parameters Δ22, Δ21 and Δ20 for the case when only two errors occur are cofactors in equation (1) for Δ33. These cofactors are:
[0039] In FIG. 2, the computations for Δ22, Δ21 and Δ20 are shown as the interim byproducts within the computations for Δ33. Similarly, Δ11 and Δ10 are cofactors in equation (48) for Δ22 which are given by
[0040] It is shown later in the specification how the equations for developing locator parameters are derived from the following prior art relationship of the error locator polynomial with the syndromes.
[0041] FIG. 3 illustrates the logic for selecting the coefficients of the error locator polynomial from the locator parameters Δ33 through Δ30 and the cofactors L22 through Δ10. The FIG. 3 logic functions to identify the number of errors from the input parameters Δ33 through Δ30 and the cofactors Δ22 through Δ10 and select the appropriate value Δm in the general equation
[0042] • When Δ33 is non-zero, indicating the presence of three errors, the coefficients Δ3 through Δ0 will assume the values Δ33 through Δ30. As shown, when Δ33 is non-zero, the output of AND-gate 41 is low, permitting the Δ32, Δ31 and A30 signals to be gated through AND-gates 42, 43 and 44 respectively since the output of AND-gate 41 is inverted at the input to each gate 42-44 to enable each of the AND-gates.
[0043] A similar logic function is achieved by Δ22 if Δ33 is zero indicating not more than two errors are present. In such a situation, Δ2, Δ1 and Δ0 will take the values of Δ22, Δ21 and Δ20 respectively through the operation of AND-gates 51, 52 and 53 respectively. This function corresponds to the syndrome equation for two errors.
[0044] The logic circuitry of FIG. 3 functions similarly - if Δ22 is also zero to cause Δ1 and Δ0 to assume the values of Δ11 and Δ10. AND-gates 61 and 62 gate Δ11 and Δ10 respectively through OR-gates 71, 72 if Δ33 and Δ22 are both zero since AND-gate 60 provides the enabling signal.
[0045] The overall logic of FIG. 3, therefore, functions to produce or to select the correct values of the coefficients Δ3, Δ2, Δ1 and Δ0 for the error locator equations from the locator parameters developed by the logic of FIG. 2.
[0046] The hardware for developing the coefficients Φ of the error value equation which is expressed as follows
will now be described. The error value equation for the generic case is developed later in the specification. The general equation can be converted to the specific three byte error correcting system set forth in the drawings such that the coefficients may be defined as
[0047] The logic circuitry of FIG. 4 in effect implements the above three equations and comprises two basic logic blocks 110 and 111. Block 110 functions to provide the product function of the two 8-bit vectors and is identical to block 10 described earlier in connection with FIG. 2. Block 111 is a sum function of two 8-bit vectors and is identical to block 11 described earlier in FIG. 2. With the error value coefficients W and the error location coefficients A being developed, the circuitry of FIG. 5 is employed to mechanize the trial and error systematic search, i.e., the Chien Search, for the actual error locations and error patterns. The circuit of FIG. 5 functions to implement the error value equation when that equation is rewritten as
[0048] As shown, FIG. 5 comprises four feedback shift registers 120-0, 120-1, 120-2 and 120-3 which receive the error location coefficients and three feedback shift. registers 130-0, 130-1 and 130-2 which receive the three error pattern coefficients. The block labelled 120 implements the following equation
while block 130 implements the numerator of equation 16. The denominator of equation 16 is developed by the inverse block 140 from the output of exclusive-OR gate 125 which is one term developed for equation 17 The function and details of the inverse block 140 are discussed further on in the specification. The output of the exclusive-OR gate 127 functions through AND-gate 128 to provide an indication of an uncorrectable error condition which, depending on the length of the codeword, would occur in some cases if the number of errors exceeds the designed correction capability of the code.
[0049] The product block 145 multiples the two 8-bit vectors in a manner identical to product blocks 110 and 10 described earlier and comprises 76 exclusive-OR gates and 64 AND-gates, each of which are 2-input gates.
[0050] The output of product brock 145 is the error pattern E. supplied to AND-block 146 comprising eight 2-input AND-gates. The error pattern vector Ei supplied to gate 146 is supplied to exclusive-OR block 147 only when the error locator logic indicates that the output of exclusive-OR block 127 is zero. The other input to exclusive-OR block 147 is from buffer 4.
[0051] As shown in FIG. 5, a clock signal C is supplied to the shift registers and buffers so that the bytes of data in error are each supplied to exclusive-OR block 147 at the appropriate time determined by the recognition that the error is located as indicated by a zero at the output of exclusive-OR block 127, thereby signifying that the output of the product block 145 contains the correct error pattern for that byte position.
[0052] The operation of the FIG. 5 circuitry is as follows. The computed values of the coefficients Δ3, Δ2, Δ1 and Δ0 and Φ2 , Φ1 and Φ0 are entered into the appropriate shift registers at clock 0 time. Each clock cycle generates a shifting operation of these registers. A shifting operation multiplies the contents of each register by a specific constant, namely α3, a2 and a , in the case of the shift
[0053] registers for Δ3, Δ2, and Δ1 respectively and α2 and a in the case of registers of Φ2 and Φ1 respectively. At the ith clock cycle, the upper set of exclusive-OR circuits 121, 125 and 127 in FIG. 5 at the output of shift registers 120 are presented with all the terms of equation 17. When the sum is 0, equation 17 is satisfied and one error location is identified. Similarly, at the ith clock cycle, the lower set of exclusive-OR circuits 131, 132 at the output of shift registers 130 are presented with all the terms of the numerator in equation 16. The denominator for equation 16 is readily available, as mentioned previously, from exclusive-OR circuit 125. Block 140 for an inverse operation and block 146 for the product . operation compute the error pattern E for each position in accordance with equation 16.
[0054] The algebraic inverse in GF(28) can be obtained through combinational logic which maps each 8-digit binary sequence into a specific 8-digit binary sequence. This mapping requires, at the most, 304 two-terminal AND-gates and 494 two-terminal OR-gates. The method of developing the inverse is shown later on in the specification. When the error location is identified, data character from buffer 4 is modified by Ei through the output-sum network 146-147. For all other values of i, the computed value of Ei is ignored since AND-gate 146 is closed. When all bytes of the codeword are delivered (at the final clock cycle c), if an error location was not identified in spite of nonzero syndromes, then there are too many errors. The latch 156 and AND-gate 157 indicates this with an UE (uncorrectable error) signal.
[0055] The corrected bytes in the decoder of FIG. 5 are delivered in the order B0, B1, B2, ..., Bc-1. This is the reverse order compared to that in the encoding operation since the check bytes correspond to the low-order positions. This reversal can easily be removed by introducing a reversal relationship between clock cycle count j and the byte location number i by substituting (c-j) for i in the decoding equations 17 and 16 and rewriting them as
[0056] In these equations, j represents the clock cycle count, and j=l to c, successively, correspond to the byte-position values i = (c-1) to 0. This provides delivery of bytes in the order Bc-1, ..., B1, B0' which is the same as that in the encoding process.
[0057] To accomplish the above-mentioned modification, the following chages are made in the logic of FIG. 5: (1) the shift register multipliers α3, α2, and a are replaced by a a and α-1, respectively; (2) the coefficients δ3, δ2, δ1, Φ2, and Φ1 are premultiplied by the constants α, α2c, αc, α2c, and αc respectively. These premultiplication circuits depend on the value of c and each requires a small number of exclusive-OR gates. Note that when c = 255, this premultiplication is not needed since α255 = 1. The modified logic described above appears in FIG. 6.
[0058] The following is the mathematical derviation for the equations used in the logic implementation of FIGS. 2 and 3.
[0059] In the three-error correcting Reed-Solomon Code in GF(28) there are six check symbols corresponding to the roots α0, α1, a , α3,α4, α5 of the generator polynomial. The corresponding syndromes are denoted by S0, S1, S2, S3, S4 and S5 respectively.
[0060] We assume that, at the most, three symbols are in error. The error values.are denoted by Eil, Ei2 and Ei3 and the locations of erroneous symbols are denoted by i1, i2 and i3. Then the relationships between the syndromes and the errors are given by
[0061] Consider the polynomial with roots at αi1, αi2 and αi3. This is called error locator polynomial, given by
Substituting x = αi in (2A) we get
[0062] From equations (lA) and (3A), we can derive the following relationship between the syndromes S. and the coefficients σi of the error location polynomial.
[0063] We can solve Equation (4A) and obtain σ0, σ1 and σ2 as
where Δ33, Δ32, Δ31 and Δ30 are given by
[0064] If the value of Δ33 is 0, then Equation (4A) is a dependent set which implies that there are fewer than three errors. In that case, the syndromes will be processed for two errors where the parameters Δ22, Δ21 and Δ20 are derived from similar equations for the case of two errors and are given by
[0066] Thus, the values Δ22' Δ21 and Δ20 for the case of two errors need not be computed separately. They are available as byproducts of the computation for Δ33. Similarly, Δ11 and Δ10 for the case of one error are given by
which are cofactors of Δ22 and are also readily available as syndromes.
[0068] The exact number of errors is determined as follows:
The special cases of two and one errors can be accomodated automatically by selecting appropriate determinants. To this end, let Δ3, Δ2, Δ1 and Δ0 be defined as
[0070] Conventionally, the error locator polynomial (3A) with the coefficients σ0, σ1 and σ2 of equation (21A) is used to determine error locations through the well known Chien search procedure. However, the error locator equation can be modified in order to avoid the division by Δv. The modified error locator equation is given as
[0072] Next an expression is derived for the error values Ei1, Ei2 and Ei3. From Equations (lA) for j = 0, 1 and 2
[0077]
Notice that the error value Ei1 in Equation (28A) is expressed in terms of i1 and hence it can be computed without explicit values for i2 and i3. The other error values Ei2 and Ei3 have similar expressions. A more general expression for error value is obtained by rewriting Equation (28A) with a running variable i in place of i1 as follows:
where the coefficients Φ0, Φ1 and Φ2 are given by
[0078] The Equation (29A) makes it possible to combine the conventional Step 4 with Step 3 of the decoding process where the error values Ei1. Ei2 and Ei3 are computed in synchronism with the "Chien Search" of the error locations.
[0079] The syndrome decoder then consists of computation of various quantities in Equations (22A) and (29A) for each value of i in a cyclic iterative manner and then affecting the correction of an outgoing symbol with E whenever Equation (22A) is satisfied.
The General Case of t Errors
[0080] The following is the mathematical derivation for the general case to establish that the logic set forth in FIGS. 2, 3, 4 and 5 for the special case of up to three errors is applicable in general for t errors.
[0081] In a general BCH or Reed-Solomon code, the codeword consists of n-symbols which include r check symbols corresponding to the roots αa, αa+1, αa+2, ..., αa+r-1 of the generator polynomial where a is an element of the Galois field GF(256). The integer a will be taken to be zero, although all of the following results can be derived with any value of a. The corresponding syndromes are denoted by S0, S1, S2, ..., Sr-1 respectively. The syndromes can be computed from the received codeword as
where B0, B1' B2' ..., Bn-1 are the n-symbols of the received codeword.
[0082] Let v denote the actual number of symbols in error in a given codeword. The error values are denoted by Ei where i represents an error location value from a set of v different error locations given by {I} = {i1, i2, ..., iv}. The relationship between syndromes and the errors are then given by
[0083] Any non-zero value of a syndrome indicates the presence of errors. The decoder processes these syndromes in order to 'determine the locations and values of the errors. Let t denote the maximum number of errors that can be decoded without ambiguity. A set of r=2t syndromes are required to determine the locations and values of t errors.
[0084] Consider the polynomial with roots at a where iε{i}. This is called the error locator polynomial defined as
where σ0 = 1, σv ≠ 0 and for m >v, σm = 0 . The unknown coefficients σm for m ≤ v can be determined from the syndromes of Equation 1B as shown below.
[0086] Using Equations 2B and 4B , it is easy to show that the syndromes S and the coefficients σm of the error locator polynomial satisfies the following set of relationships:
[0088] Let M denote the tx(t+l) syndrome matrix on left side of Equation 6B. Let Mt denote the square matrix obtained by eliminating the last column in matrix M. If Mt is nonsingular, then the above set of equations can be solved using Cramer's rule to obtain
where Δtt is the non-zero determinant of matrix Mt and Δtm denotes the determinant of the matrix obtained by replacing the mth column in matrix Mt by negative of the last column of the syndrome matrix M for each m = 0, 1, ..., t-1.
[0089] If matrix Mt is singular, (i.e., Δtt is 0) then Equation 5B is a dependent set which implies that there are fewer than t errors. In that case, σt is 0. We can delete σt and last row and last column of the syndrome matrix in 6B. The resulting matrix equation corresponds to that for t-1 errors. This process is repeated if necessary so that the final matrix equation corresponds to that for v errors and M is nonsingular. Then we need the set of determinants A where m=0, 1, ..., v.
[0090] It can be easily seen that Δvm for v=t-l is a cofactor of Δtt corresponding to the (m-1)st column and tth row - in matrix Mt. We can express Δtt in terms of these cofactors:
Thus the values Δvm for v=t-1 need not require separate computations. They are available as byproduct of the computation for Δtt. In fact, Δvm for subsequent smaller values of v are all available as byproducts of the computation for Δtt through the hierarchical relationships of lower order cofactors. Thus, in case of fewer errors, the decoder finds Δtt =0 and automatically back tracks into prior computations to the correct value for v and uses the already computed cofactors Δvm. This is illustrated previously through hardware implmentation of the case t=3..
[0091] In order to accomodate the special cases of all fewer errors, we will replace Equation 7B by a more convenient general form
where v is determined from the fact that Δmm=0 for all m>v and A ≠0; and A is defined with the new notation vv m as
[0092] Since σ0=1 , one can determine σm for all values of m using Equation 9B. However, we will see that the coefficients σm are not needed in the entire decoding process. To this end, we obtain a modified error locator equation from Equations 4B and 9B as given by
[0093] The error location values iE{I} are the set of v unique values of i which satisfy Equation 11B.
[0094] The error locator polynomial as defined by Equation 2B has v roots corresponding to v error location values. Consider a polynomial which has all roots of the error locator polynomial except one corresponding to the location value i=j. This polynomial is defined as
[0095] When the actual number of error locations v is less than t, the coefficients σj,m are 0 for m=v,...,t-1. This is done in order to affect processing of all values of v through a single set of hardware. Substituting x=αi in Equation 12B we get
[0096] Examine an expression similar to one in Equation 5B involving the syndromes S. and the coefficients a of the new polynomial. Using Equation 2B , we substitute for Sm and get
[0100] The above expression for error values is well-known. We will reduce it further to obtain a more convenient form. To this end, we prove the following Lemmas 1, 2 and 3.
[0101] In Lemma 1, we obtain a relation which expresses the coefficients σj,m in terms of the known coefficients σk of the error locator polynomial.
Lemma 1
[0104] Comparing the coefficients of each term in the polynomials on the two sides of equation 19B we have
[0108] Next we rewrite the denominator of the expression in Equation 17B using the result of Lemma 1.
Lemma 2
[0113] Now we use the result of Lemma 1 and obtain a more convenient expression for the numerator in Equation 17B in the following Lemma.
Lemma 3
[0115] Proof: Using Lemma 1 we can express σj,m in terms of σk. On substituting these values we obtain
[0116] Let RHS denote the right-hand side of Equation (28B) In order to prove the Lemma, we rearrange RHS. First, we use the result from Equation 4B and rewrite some of the terms in RHS (the terms with m>µ) as
Corollary:
[0126] Lemma 3 and its corollary provides a family of expressions for the numerator in Equation 17B through a choice of values for µ. We make the following observations:
(a) The expression in Lemma 3 requires the syndromes Sµ+0, Su+1, ..., Sµ+t-1; whereas the expression in the corollary requires the syndromes S0, S1, ..., St-1. This is especially important in the case of erasure correction.
(b) The expression. in the corollary requires fewer number of multiplications of the type σkSk-m compared to the expression in Lemma 3. In. the corollary, this number depends on the choice of µ and is the minimum when u=µ where u is the largest integer under t/2. The number of multiplications with the choice µ=µ is the integer closest to (t2+2t)/4.
(c) The cases of fewer errors (v<t) are automatically processed by the same hardware as that for processing t errors. With a given choice of µ, the same hardware can be also used in applications with a smaller set of syndromes, S0, S1, ..., St1 and up to tl errors provided that t≥t1>µ. From this point .of view, lower values of µ are desirable and the choice of µ=0 offers the maximum flexibility in applicability of the decoder hardware. The number of multiplications required by the expression in the corollary with µ=0 is (t2-t+2)/2 which is less than two times the minimum of (t2+2t)/4 when u=µ.
[0127] In view of the above observations, we will use expression in the corollary of Lemma 3 in decoder implementation. For large values of t, it is advisable to use u=µ for the greatest economy in hardware. In case of small values of t, we use u=0 for flexibility in being able to reduce the value of t, later if necessary, without modifying the already fabricated hardware.
[0128] Now we can rewrite Equation 17B using the results of Lemma 2 and the corollary of Lemma 3 with µ=0. As a result, any error value E. can be expressed as
where
[0129] In view of Equation 9A, the above expressions can be normalized to obtain error values in terms of A as follows:
where
[0130] In case of binary base field, the terms with even values of m, (m mod 2=0) in the denomination of Equation 39B vanish. The resultant expression for E. for binary base field is
where
[0131] Note that the computations for the denominator in Equation 41B is already available as a byproduct from the computations of Equation 11B for each value of i in the Chien Search of error locations. The numerator for each value of i can be computed and multiplied by the inverse of the denominator in synchronism with the search for error locations. The resultant E. is used for correction of the outgoing ith data symbol Bi whenever the error locator Equation 11B is satisfied.
[0132] Table 2 which follows lists 255 non-zero elements of GF(28), each with the corresponding inverse element. A primitive polynomial p(x) = x8 + x7 + x5 + x3 + 1 was used to generate these elements. The elements are represented by 8-digit binary vectors which in polynomial notation have the coefficient for the high order term on the left. The inverse function of this table was implemented through block 40 using the conventional 8-bit encode-decode logic. This requires at the most 304 AND-gates and 494 OR-gates.
[0133] While the invention has been particularly shown and described with reference to a preferred embodiment thereof, it will be understood by those skilled in the art that various other changes in the form and details may be made therein without departing from the spirit and scope of the invention.
1. A multibyte error correcting system operable to detect up to t errors in a codeword
containing up to 2 b character positions where each character is represented by a
byte consisting of a unique combination of "b" binary bits by processing 2t syndrome
bytes, each containing "b" bits which were generated by combining write check bytes
and read check bytes that correspond to the modulo 2 sum of bit positions of said
code word that were selected systematically in accordance with a parity check matrix
reflecting the roots αa,αa+1, αa+2,...,αa+2t-1 of the generator polynomial where α is an element of the finite field of 2b elements, said system including:
syndrome generating means for generating 2t syndrome bytes from reading one code word into said system;
clocking means for controlling the serial transfer of each byte of said one codeword from said ECC system and each byte of the next codeword into said ECC system during each clock cycle; and
means responsive to said 2t syndrome bytes for correcting during said serial byte transfer from said system up to t erroneous bytes in said one codeword as said other codeword is entered into said ECC system whereby said one codeword and said next codeword are processed in a continuous stream of serial bytes.
2. The error correcting system defined in claim 1 further including a buffer for storing
bytes of said codewords during said transfer to and from said system.
3. The system recited in claim 2 in which said buffer is operable to receive up to
2b bytes of information where each byte is defined by b bits.
4. The system recited in claim 3 in which said means for correcting comprises combinatorial
logic circuitry for generating t+1 coefficients for the error locator polynomial having
t roots which indicate the character positions in said codeword containing said t
errors.
5. The system recited in claim 4 in which said logic circuitry is operable to generate
t+1 coefficients from 2t syndrome bytes to permit correcting up to t errors wherein
said logic circuitry operates to identify one character position in said codewords
for each error up to t errors.
6. The system recited in claim 5 in which said logic circuitry comprises a plurality
of product blocks and a plurality of exclusive-OR blocks, each of said product blocks
comprising a plurality of two input AND-gates and a plurality of two input exclusive-OR
gates, each of said product blocks providing a "b" bit product output signal from
two "b" bit input signals corresponding to said preselected syndrome bytes.
7. The system recited in claim 6 further including means for combining said product
output signals from said product blocks to provide error parameter signals from which
said t+1 coefficients of said error locator polynomial are selected.
8. The system recited in claim 7 further including means for selecting from said error
parameter signals, said coefficients of said error locator polynomial in accordance
with the number of errors in said codeword.
9. The system recited in claim 8 in which said means for selecting includes logic
circuitry for indicating the number of said errors.
10. The system recited in claim 4 further including means for developing t error value
coefficients ? from t different preselected syndrome bytes and t different error locator
coefficients Δ.
11. The system recited in claim 10 in which said means for correcting further includes
a plurality of feedback shift registers, and means for supplying each said error value
coefficient $ to a separate said register and means for supplying each said error
locator coefficient to a separate said shift register.
12. The system recited in claim 11 in which said clocking means includes a clock pulse
generator for supplying a clock pulse to each of said shift registers, each said shift
register having a predetermined feedback pattern to cause said coefficients supplied
thereto to be shifted a predetermined number of element positions in said 2b finite field each time a clock pulse is supplied to said register, first means for
combining the output of said shift registers associated with said error locator coefficients
a , and means for detecting when the output of said Δ coefficient combining means
corresponds to a predetermined element of said finite field.
13. The system recited in claim 12 further including second means for combining the
output of selected said shift registers associated with said error value coefficients
φ and means for combining the output of said second combining means with the inverse
of the output of selected said registers associated with said error locator coefficients
A in accordance with said error value equation to produce an error value during the
clock interval that said first combining means provides said output corresponding
to said predetermined element of said finite field.
14. A multibyte error correcting method for correcting up to t errors in a codeword
containing up to 2b character positions where each character is represented by a byte consisting of a
unique combination of "b" binary bits by processing 2t syndrome bytes, each containing
"b" bits which were generated by combining write check bytes and read check bytes
that correspond to the modulo 2 sum of bit positions of said code word that were selected
systematically in accordance with a parity check matrix reflecting the roots αa,αa+1 αa+2,..., αa+2t-1 of the generator polynomial where α is an element of the finite field of 2b elements, said method comprising:
generating 2t syndrome bytes by reading one codeword into said system;
controlling the serial transfer of bytes of said one codeword from said ECC system and bytes of the next codeword into said ECC system by a clock signal so that during each clock cycle one byte of said next codeword is transferred into said system and one byte of said one one codeword is transferred out of said system; and
correcting up to t erroneous bytes in said one codeword in response to said 2t syndrome bytes during the transfer of bytes from said system as said other codeword is entered into said ECC system whereby said one codeword and said next codeword are processed in a continuous stream of serial bytes.
15. The method recited in claim 14 in which said step of correcting is characterized
by at least one byte of said codeword being transferred from said system prior to
identifying all byte positions of said codeword containing errors.